The orthogonal group over a local ring is \(4\)-reflectional. (English) Zbl 0801.20025
Let \(R\) be a commutative local ring such that 2 is a unit in \(R\). The author shows that every element in the orthogonal group of a free \(R\)- module of finite rank is a product of two involutions in that group. For a cyclic isometry the involutions are constructed directly. This, combined with the bireflectionality of an orthogonal group for a vector space, produces the desired result.
Reviewer: E.Ellers (Toronto)
MSC:
| 20H25 | Other matrix groups over rings |
| 15B33 | Matrices over special rings (quaternions, finite fields, etc.) |
| 20F05 | Generators, relations, and presentations of groups |
| 20G35 | Linear algebraic groups over adèles and other rings and schemes |
Keywords:
product of involutions; commutative local ring; orthogonal group; free \(R\)-module of finite rank; cyclic isometry; bireflectionalityReferences:
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